I plotted this $f(s)$ before. It might be easier to visualize the roots of $|f(s)|=1$ from the contour plot below.
Milgram (arxiv:0911.1332v2) noticed the roots of $|f(s)|=1$ with $\Re(s)\not=1/2$.
Spira (1965,AN INEQUALITY FOR THE RIEMANN ZETA FUNCTION) proved that:
THEOREM 1. For $y\geq 10, 1/2 <x< 1, |f(s)|> 1$.
The lower bound for $y$ has been reduced to a number around 7 by others (cf. the papers that cited Spira's paper. It is also mentioned somewhere in these papers that people are wondering if the Riemann hypothesis can be proved only using the knowledge of the functional equation $ζ(s)=f(s)ζ(1-s)$).
But Spira mentioned that his theorem can not be directly used to prove that $\zeta(s)$ does not have zeros off the critical line ($\Re(s)=1/2$) for $y\geq 10$.
Albeverio and Cebulla (2007,Müntz formula and zero free regions for the Riemann
zeta function, Bull. Sci. math. 131 (2007) 12–38) proved that
Corollary 4.10. There are no zeros of the $\zeta(s)$ function of the form $s = x + iy$, $x \in (0, 1)$, $0< |y|< 2\pi/\log(2)\approx9.06$.
If one can use a variation of Spira's theorem 1 to prove that $\zeta(s)$ does not have zeros off the critical line for $y\geq 9$, then, in view of Albeverio and Cebulla's Corollary 4.10., one can prove that $\zeta(s)$ does not have zeros off the critical line ($\Re(s)=1/2$) for $0<|y|$.